Translation-invariant linear forms on \({\mathcal E}'(G)\) and non-Liouville tuples on G
Let G be a locally compact, compactly generated Abelian group; let \({\mathcal E}'(G)\) be the Schwartz-Bruhat space of distributions on G having compact supports. Then every translation-invariant linear form on \({\mathcal E}'(G)\) is continuous if and only if G contains a finite ``non-Liouville'' set. To understand this, one must have a definition of ``non-Liouville set''; this, like the theorem itself, is a generalization from the case of the torus group. If \(x\in {\mathbb{R}}\), let \(x\) denote the minimal distance from x to an integer. Let \(\Pi\) denote the canonical map from \({\mathbb{R}}^ m\) onto \({\mathbb{T}}^ m={\mathbb{R}}^ m/{\mathbb{Z}}^ m\). A subset \(\{t_ 1,...,t_ n\}\) of \({\mathbb{T}}^ m\) is a non-Liouville n- tuple if and only if there are real numbers C, \(b>0\) such that, for any \(a_{\nu}\) satisfying \(\Pi (a_{\nu})=t_{\nu}\), and any nonzero \(k\in {\mathbb{Z}}^ m\), \(\sum^{n}_{\nu =1}[[ <k,a_{\nu}>]]\geq C(1+| k|)^{-b}\). An equivalent condition is: for every character \(\chi\) of the group \({\mathbb{T}}^ m\), not identically equal to 1, \(\sum^{n}_{\nu =1}| 1-\chi (a_{\nu})| \geq C(1+| k|)^{-b}\) (for a possibly different value of C). The latter inequality is easily generalized from \({\mathbb{T}}^ m\) to an arbitrary compact Abelian group G and its dual. If G is a locally compact Abelian group, then a finite subset \(\{a_ 1,...,a_ n\}\) is a non-Liouville n-tuple iff there is a discrete, finitely generated subgroup H of G such that G/H is compact and the images of \(a_ 1,...,a_ n\) are a non-Liouville n-tuple in G/H.
locally compact, compactly generated Abelian group
scientific article; zbMATH DE number 4060049
scientific article